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Chapter 7: Problem 27

Particle A is at the origin and has a mass of \(30.0 \mathrm{g} .\) Particle B has a mass of \(10.0 \mathrm{g} .\) Where must particle \(\mathrm{B}\) be located if the coordinates of the \(\mathrm{cm}\) are \((x, y)=(2.0 \mathrm{cm},\) $5.0 \mathrm{cm}) ?$

Short Answer

Expert verified
Answer: The coordinates of Particle B are (8.0 cm, 20.0 cm).

Step by step solution

01

Write down the formula for the center of mass in two dimensions

The formula for the center of mass in two dimensions is given by: \(x_{cm} = \frac{m_Ax_A + m_Bx_B}{m_A + m_B}\) \(y_{cm} = \frac{m_Ay_A + m_By_B}{m_A + m_B}\) where \(m_A\) and \(m_B\) are the masses of particles A and B, respectively, and \((x_A, y_A)\) and \((x_B, y_B)\) are the coordinates of particles A and B, respectively.
02

Substitute the given values into the x-coordinate formula for center of mass

We are given \(x_{cm} = 2.0 cm\), \(m_A = 30.0 g\), \(x_A = 0\), and \(m_B = 10.0 g\). Plugging these values into the x-coordinate formula, we get: \(2.0 = \frac{30.0 \times 0 + 10.0 \times x_B}{30.0 + 10.0}\)
03

Solve for the x-coordinate of Particle B

Simplifying and solving for \(x_B\), we obtain: \(2.0 = \frac{10.0x_B}{40.0}\) \(x_B = \frac{2.0 \times 40.0}{10.0}\) \(x_B = 8.0 \;\mathrm{cm}\)
04

Substitute the given values into the y-coordinate formula for center of mass

We are given \(y_{cm} = 5.0 cm\), \(m_A = 30.0 g\), \(y_A = 0\), and \(m_B = 10.0 g\). Plugging these values into the y-coordinate formula, we get: \(5.0 = \frac{30.0 \times 0 + 10.0 \times y_B}{30.0 + 10.0}\)
05

Solve for the y-coordinate of Particle B

Simplifying and solving for \(y_B\), we obtain: \(5.0 = \frac{10.0y_B}{40.0}\) \(y_B = \frac{5.0 \times 40.0}{10.0}\) \(y_B = 20.0 \;\mathrm{cm}\)
06

Combine x and y coordinates for Particle B

Now that we have found both the x and y coordinates of Particle B, we can combine them to find its location: The location of Particle B is \((8.0 \;\mathrm{cm}, 20.0 \;\mathrm{cm})\).

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Most popular questions from this chapter

An object located at the origin and having mass \(M\) explodes into three pieces having masses \(M / 4, M / 3,\) and \(5 M / 12 .\) The pieces scatter on a horizontal frictionless xy-plane. The piece with mass \(M / 4\) flies away with velocity \(5.0 \mathrm{m} / \mathrm{s}\) at \(37^{\circ}\) above the \(x\)-axis. The piece with mass \(M / 3\) has velocity \(4.0 \mathrm{m} / \mathrm{s}\) directed at an angle of \(45^{\circ}\) above the \(-x\)-axis. (a) What are the velocity components of the third piece? (b) Describe the motion of the CM of the system after the explosion.
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